4-1 Chapter 4 Measures of distance between samples: Euclidean We will be talking a lot about distances in this book. The concept of distance between two samples or between two variables is fundamental in multivariate analysis – almost everything we do has a relation with this measure. If we talk about a single variable we take this concept for granted. If one sample has a pH of 6.1 and another a pH of 7.5, the distance between them is 1.4: but we would usually call this the absolute difference. But on the pH line, the values 6.1 and 7.5 are at a distance apart of 1.4 units, and this is how we want to start thinking about data: points on a line, points in a plane, … even points in a ten- dimensional space! So, given samples with not one measurement on them but several, how do we define distance between them. There are a multitude of answers to this question, and we devote three chapters to this topic. In the present chapter we consider what are called Euclidean distances, which coincide with our most basic physical idea of distance, but generalized to multidimensional points. Contents Pythagoras’ theorem Euclidean distance Standardized Euclidean distance Weighted Euclidean distance Distances for count data Chi-square distance Distances for categorical data Pythagoras’ theorem The photo shows Michael in July 2008 in the town of Pythagorion, Samos island, Greece, paying homage to the one who is reputed to have made almost all the content of this book possible: ΠΥΘΑΓΟΡΑΣ Ο ΣΑΜΙΟΣ, Pythagoras the Samian. The illustrative geometric proof of Pythagoras’ theorem stands carved on the marble base of the statue – it is this theorem that is at the heart of most of the multivariate analysis presented in this book, and particularly the graphical approach to data analysis that we are strongly promoting. When you see the word “square” mentioned in a statistical text (for example, chi square or least squares), you can be almost sure that the corresponding theory has some relation to this theorem. We first show the theorem in its simplest and most familiar two-dimensional form, before showing how easy it is to generalize it to multidimensional space. In a right- 4-2 angled triangle, the square on the hypotenuse (the side denoted by A in Exhibit 4.1) is equal to the sum of the squares on the other two sides (B and C); that is, A 2 = B 2 + C 2. Exhibit 4.1 Pythagoras’ theorem in the familiar right-angled triangle, and the monument to this triangle in the port of Pythagorion, Samos island, Greece, with Pythagoras himself forming one of the sides. 2 2 2 A = B + C B A C Euclidean distance The immediate consequence of this is that the squared length of a vector x = [ x1 x2 ] is the sum of the squares of its coordinates (see triangle OPA in Exhibit 4.2, or triangle OPB – |OP| 2 denotes the squared length of x, that is the distance between point O and P); and the Exhibit 4.2 Pythagoras’ theorem applied to distances in two-dimensional space. Axis 2 2 = 2 + 2 2 = − 2 + − 2 | OP | x1 x2 | PQ | (x1 y1) (x2 y2 ) P B • x = [ x x ] x2° 1 2 |x –y | 2 2 Q y • y = [ y y ] 2 D 1 2 A O ° Axis 1 x1 y1 |x 1 –y1| 4-3 squared distance between two vectors x = [ x1 x2 ] and y = [ y1 y2 ] is the sum of squared differences in their coordinates (see triangle PQD in Exhibit 4.2; |PQ| 2 denotes the squared distance between points P and Q). To denote the distance between vectors x and y we can use the notation dx,y so that this last result can be written as: 2 2 2 dx,y = ( x1 – y1) + ( x2 – y2) (4.1) that is, the distance itself is the square root = − 2 + − 2 d x,y (x1 y1 ) (x2 y2 ) (4.2) What we called the squared length of x, the distance between points P and O in Exhibit 4.2, is the distance between the vector x = [ x1 x2 ] and the zero vector 0 = [ 0 0 ] with coordinates all zero: = 2 + 2 d x,0 x1 x2 (4.3) which we could just denote by dx . The zero vector is called the origin of the space. Exhibit 4.3 Pythagoras’ theorem extended into three dimensional space Axis 3 C x3 ° x = [ x1 x2 x3 ] • P 2 = 2 + 2 + 2 | OP | x1 x2 x3 x2 O ° Axis 2 B A x1 ° °S Axis 1 4-4 We move immediately to a three-dimensional point x = [ x1 x2 x3 ], shown in Exhibit 4.3. This figure has to be imagined in a room where the origin O is at the corner – to reinforce this idea ‘floor tiles’ have been drawn on the plane of axes 1 and 2, which is the ‘floor’ of the room. The three coordinates are at points A, B and C along the axes, and the angles AOB, AOC and COB are all 90 ° as well as the angle OSP at S, where the point P (depicting vector x) is projected onto the ‘floor’. Using Pythagoras’ theorem twice we have: |OP| 2 = |OS| 2 + |PS| 2 (because of right-angle at S) |OS| 2 = |OA| 2 + |AS| 2 (because of right-angle at A) and so |OP| 2 = |OA| 2 + |AS| 2 + |PS| 2 that is, the squared length of x is the sum of its three squared coordinates and so = 2 + 2 + 2 d x x1 x2 x3 It is also clear that placing a point Q in Exhibit 4.3 to depict another vector y and going through the motions to calculate the distance between x and y will lead to = − 2 + − 2 + − 2 d x,y (x1 y1 ) (x2 y2 ) (x3 y3 ) (4.4) Furthermore, we can carry on like this into 4 or more dimensions, in general J dimensions, where J is the number of variables. Although we cannot draw the geometry any more, we can express the distance between two J-dimensional vectors x and y as: J = − 2 dx,y ∑(x j y j ) (4.5) j=1 This well-known distance measure, which generalizes our notion of physical distance in two- or three-dimensional space to multidimensional space, is called the Euclidean distance (but often referred to as the ‘Pythagorean distance’ as well). Standardized Euclidean distance Let us consider measuring the distances between our 30 samples in Exhibit 1.1, using just the three continuous variables pollution, depth and temperature. What would happen if we applied formula (4.4) to measure distance between the last two samples, s29 and s30 , for example? Here is the calculation: = − 2 + − 2 + − 2 = + + = ds29, s30 0.6( )9.1 51( 99 ) 0.3( )9.2 16 81. 2304 01.0 2320 82. = 48.17 4-5 The contribution of the second variable depth to this calculation is huge – one could say that the distance is practically just the absolute difference in the depth values (equal to |51-99| = 48) with only tiny additional contributions from pollution and temperature. This is the problem of standardization discussed in Chapter 3 – the three variables are on completely different scales of measurement and the larger depth values have larger inter- sample differences, so they will dominate in the calculation of Euclidean distances. Some form of standardization is necessary to balance out the contributions, and the conventional way to do this is to transform the variables so they all have the same variance of 1. At the same time we centre the variables at their means – this centring is not necessary for calculating distance, but it makes the variables all have mean zero and thus easier to compare. The transformation commonly called standardization is thus as follows: standardized value = (original value – mean) / standard deviation (4.5) The means and standard deviations of the three variables are: Pollution Depth Temperature mean 4.517 74.433 3.057 s.d. 2.141 15.615 0.281 leading to the table of standardized values given in Exhibit 4.4. These values are now on Exhibit 4.4 Standardized values of the three continuous variables of Exhibit 1.1 SITE ENVIRONMENTAL VARIABLES NO. Pollution Depth Temperature s1 0.132 -0.156 1.576 s2 -0.802 0.036 -1.979 s3 0.413 -0.988 -1.268 s4 1.720 -0.668 -0.557 s5 -0.288 -0.860 0.154 s6 -0.895 1.253 1.576 s7 0.039 -1.373 -0.557 s8 0.272 -0.860 0.865 s9 -0.288 -0.412 1.221 s10 2.561 -0.348 -0.201 s11 0.926 -1.116 0.865 s12 -0.335 0.613 0.154 s13 2.281 -1.373 -0.201 s14 0.086 0.549 -1.979 s15 1.020 1.637 -0.913 s16 -0.802 0.613 -0.201 s17 0.880 1.381 0.154 s18 -0.054 -0.028 -0.913 s19 -0.662 0.292 1.932 s20 0.506 -0.092 -0.201 s21 -0.101 -0.988 1.221 s22 -1.222 -1.309 -0.913 s23 -0.989 1.317 -0.557 s24 -0.101 -0.668 -0.201 s25 -1.175 1.445 -0.201 s26 -0.942 0.228 1.221 s27 -1.129 0.677 -0.201 s28 -0.522 1.125 0.865 s29 0.693 -1.501 -0.201 s30 -1.222 1.573 -0.557 4-6 comparable standardized scales, in units of standard deviation units with respect to the mean.